To compute the radiation spectrum, we now have two possible sources of gauge-dependence, one from the external field and the other from the background field. For the non-perturbative, adiabatic, and relativistic external field, we just explained in the previous Section that the gauge-choice seriously affects the energy levels of our quantum system. As we will shortly show, this indeed leads to significant gauge dependence of the spectrum. For the background field, since it does not contribute a non-perturbative scalar potential, its gauge-dependence can be handled by Eq. (REF ), which agrees well with the measured atomic spectroscopy, as Lamb et. al. elaborated in Ref. [1]}.
We can expect that this method still applies when coupling the background field to our non-perturbative time-dependent system. (Although our comments in the paragraph around Eq. (REF ) still hold here.) So, analogous to Eq. (REF ), the instantaneous eigenstate under the influence of the background electromagnetic field is defined by applying another unitary transformation:
\({}\left| n_{BG}(t) \right> =U_{BG} \left| n_{G}(t) \right> ,\) 
